Cut tips off melons with a sharp knife; cut in half lengthwise. Scoop out white interior and seeds with a spoon. Rinse the melon and cut into very small pieces. Place in a bowl and generously add kosher salt, mixing until melon is completely covered. Cover and refrigerate until flavors combine, 3 to 4 hours.

Rinse melon to remove excess salt. Place into a pot and cover with room temperature or cool water; cover and bring to a boil. Cook until melon has softened slightly, about 1 minute. Remove from heat; drain the melon.

Heat sesame oil in a large skillet over medium-high heat. Add pork; season with garlic salt and garlic pepper. Stir-fry just until the pink on the surface of the pork disappears, 3 to 4 minutes. Add melon and stir fry until soft, about 2 minutes more.

Reduce heat to medium. Push pork mixture to the edges of the skillet with a spatula. Crack eggs into the center of the skillet; scramble until firm, 3 to 5 minutes. Stir contents of the skillet together until combined. Season with soy sauce.